On the Complexity of Differentially Private Best-Arm Identification with Fixed Confidence

• URL: http://arxiv.org/abs/2309.02202v1
• Date: Tue, 5 Sep 2023 13:07:25 GMT
• Title: On the Complexity of Differentially Private Best-Arm Identification with Fixed Confidence
• Authors: Achraf Azize, Marc Jourdan, Aymen Al Marjani, Debabrota Basu
• Abstract summary: We study the problem of Best Arm Identification with fixed confidence under $epsilon$-global Differential Privacy. Our lower bound suggests the existence of two privacy regimes depending on the privacy budget. We propose AdaP-TT, an $epsilon$-global DP variant of the Top Two algorithm.
• Score: 16.295693624977563
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Best Arm Identification (BAI) problems are progressively used for data-sensitive applications, such as designing adaptive clinical trials, tuning hyper-parameters, and conducting user studies to name a few. Motivated by the data privacy concerns invoked by these applications, we study the problem of BAI with fixed confidence under $\epsilon$-global Differential Privacy (DP). First, to quantify the cost of privacy, we derive a lower bound on the sample complexity of any $\delta$-correct BAI algorithm satisfying $\epsilon$-global DP. Our lower bound suggests the existence of two privacy regimes depending on the privacy budget $\epsilon$. In the high-privacy regime (small $\epsilon$), the hardness depends on a coupled effect of privacy and a novel information-theoretic quantity, called the Total Variation Characteristic Time. In the low-privacy regime (large $\epsilon$), the sample complexity lower bound reduces to the classical non-private lower bound. Second, we propose AdaP-TT, an $\epsilon$-global DP variant of the Top Two algorithm. AdaP-TT runs in arm-dependent adaptive episodes and adds Laplace noise to ensure a good privacy-utility trade-off. We derive an asymptotic upper bound on the sample complexity of AdaP-TT that matches with the lower bound up to multiplicative constants in the high-privacy regime. Finally, we provide an experimental analysis of AdaP-TT that validates our theoretical results.

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